Mathematics

We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions 8 and 24, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions 12, 16, 20, 28, and 32. More generally, we provide a systematic technique for proving separations of this sort.

If we fix the angles at the vertices of a convex planar n -gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of n edge lengths form a convex polytope of dimension n−3 , each facet of which consists of those n -gons in which the length of a particular edge has fallen to zero. Bavard and Ghys require unit area instead, which gives them a hyperbolic polytope. Those two polytopes are combinatorially equivalent, so either is fine for our purposes. Such a fixed-angles polytope is combinatorially richer when the angles are well balanced. We say that fixed external angles are "majority dominant" when every consecutive string of more than half of them sums to more than π . When n is odd, we show that the fixed-angles polytope for any majority-dominant angles is dual to the cyclic polytope C n−3 (n) . To extend that result to even n , we require that the angles also have "dipole tie-breaking": None of the n strings of length n/2 sums to precisely π , and the n/2 that sum to more than π overlap as much as possible, all containing a particular angle. Fixing the vertex angles is uncommon, however; people more often fix the edge lengths. That is harder, in part because fixed-lengths n -gons may not be convex, but mostly because fixing the lengths constrains the angles nonlinearly -- so the resulting moduli spaces, called "polygon spaces", are curved. Using Schwarz-Christoffel maps, Kapovich and Millson show that the subset of that polygon space in which the n -gons are convex and traversed counterclockwise is homeomorphic to the fixed-angles polytope above, for those same fixed values. Each such subset is thus a topological polytope; and it is dual cyclic whenever the fixed lengths are majority dominant and, for even n , have dipole tie-breaking.

We generalize a result of Freedman and He, concerning the duality of moduli and capacities in solid tori, to sufficiently regular metric spaces. This is a continuation of the work of the author and K. Rajala on the corresponding duality in condensers.

This paper investigates how the geometry of a cycle in the loop space of a Riemannian manifold controls its topology. For fixed β∈ H n (ΩX;R) one can ask how large |⟨β,Z⟩| can be for cycles Z supported in loops of length ≤L and of volume ≤ L n−1 for a suitably defined notion of volume of in loop space. We show that an upper bound to this question provides upper bounds Gromov's distortion of higher homotopy groups. We also show that we can exhibit better lower bounds than are currently known for the corresponding questions for Gromov's distortion. Specifically, we show there exists a β detecting the homotopy class of the puncture in [( CP 2 ) #4 × S 2 ] ∘ and a family of cycles Z L with the geometric bounds above such that |⟨β,Z⟩|=Ω( L 6 /logL) .

Starting from a finite simple graph G , for each eigenvalue θ of its adjacency matrix one can construct a convex polytope P G (θ) , the so called θ -eigenpolytop of G . For some polytopes this technique can be used to reconstruct the polytopes from its edge-graph. Such polytopes (we shall call them spectral) are still badly understood. We give an overview of the literature for eigenpolytopes and spectral polytopes. We introduce a geometric condition by which to prove that a given polytope is spectral (more exactly, θ 2 -spectral). We apply this criterion to the edge-transitive polytopes. We show that every edge-transitive polytope is θ 2 -spectral, is uniquely determined by this graph, and realizes all its symmetries. We give a complete classification of distance-transitive polytopes.

Self-similar sets with open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields. .

The action of the rotation group SO(3) on systems of n points in the 3 -dimensional Euclidean space R 3 induces naturally an action of SO(3) on R 3n . In the present paper we consider the following question: do there exist 3 polynomial functions f 1 , f 2 , f 3 on R 3n such that the intersection of the set of common zeros of f 1 , f 2 , and f 3 with each orbit of SO(3) in R 3n is nonempty and finite? Questions of this kind arise when one is interested in relative motions of a given set of n points, i.e., when one wants to exclude the local motions of the system of points as a rigid body. An example is the problem of deciding whether a given polyhedron is non-trivially flexible. We prove that such functions do exist. To get a necessary system of equations f 1 =0 , f 2 =0 , f 3 =0 , we show how starting by choice of a hypersurface in CP n−1 containing no conics, no lines, and no real points one can find such a system.

We show that for any Carnot group G there exists a natural number D G such that for any 0<ε<1/2 the metric space (G, d 1−ε G ) admits a bi-Lipschitz embedding into R D G with distortion O G ( ε −1/2 ) . This is done by building on the approach of T. Tao (2018), who established the above assertion when G is the Heisenberg group using a new variant of the Nash--Moser iteration scheme combined with a new extension theorem for orthonormal vector fields. Beyond the need to overcome several technical issues that arise in the more general setting of Carnot groups, a key point where our proof departs from that of Tao is in the proof of the orthonormal vector field extension theorem, where we incorporate the Lovász local lemma and the concentration of measure phenomenon on the sphere in place of Tao's use of a quantitative homotopy argument.

Inspired by the recent theory of Entropy-Transport problems and by the D -distance of Sturm on normalised metric measure spaces, we define a new class of complete and separable distances between metric measure spaces of possibly different total mass. We provide several explicit examples of such distances, where a prominent role is played by a geodesic metric based on the Hellinger-Kantorovich distance. Moreover, we discuss some limiting cases of the theory, recovering the "pure transport" D -distance and introducing a new class of "pure entropic" distances. We also study in detail the topology induced by such Entropy-Transport metrics, showing some compactness and stability results for metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense.

This paper presents an additional class of regular polyhedra--envelope polyhedra--made of regular polygons, where the arrangement of polygons (creating a single surface) around each vertex is identical; but dihedral angles between faces need not be identical, and some of the dihedral angles are 0 degrees (i.e., some polygons are placed back to back). For example, squares, 6 around a point, is produced by deleting the triangles from the rhombicuboctahedron, creating a hollow polyhedron of genus 7 with triangular holes connecting 18 interior and 18 exterior square faces. An empty cube missing its top and bottom faces becomes an envelope polyhedron, squares, 4 around a point, with a toroidal topology. This definition leads to many interesting finite and infinite multiply connected regular polygon networks, including one infinite network with squares, 14 around a point, and another with triangles, 18 around a point. These are introduced just over 50 years after my related paper on infinite spongelike pseudopolyhedra in American Mathematical Monthly (Gott, 1967).